After finding the values by using half angle formula, how to determined the quadrant
A + or - sign appears in many of the half angle formulas. You determine which sign to use, and which quadrant the half angle is in, by knowing the value of the FULL angle, A. If A is in the first of secnd quadrant, then A/2 in in the first quadrant. If A is in the third or fourth quadrant, then A is in the second quadrant, etc.
To determine the quadrant of an angle after finding its values using the half angle formula, you need to consider the signs of the trigonometric functions in that quadrant.
The half angle formula for sine is given by:
sin(x/2) = ±√((1 - cos(x))/2)
And the half angle formula for cosine is given by:
cos(x/2) = ±√((1 + cos(x))/2)
Depending on the original angle x and the quadrant it lies in, you can determine the appropriate signs for the half angle formula.
Here's how you can determine the quadrant after finding the values using the half angle formula:
1. Determine the original angle: If the original angle x was given, determine which quadrant it lies in. This will help you understand the possible signs of the half angle.
2. Consider the signs: Take note of the signs in the half angle formulas. The ± sign indicates that the values can be either positive or negative, depending on the quadrant.
3. Analyze the quadrant: Based on the original angle x and its quadrant, examine the signs of the trigonometric functions that apply to that quadrant.
- For the first quadrant (0° < x < 90°), both sine and cosine are positive.
- For the second quadrant (90° < x < 180°), sine is positive, and cosine is negative.
- For the third quadrant (180° < x < 270°), both sine and cosine are negative.
- For the fourth quadrant (270° < x < 360°), sine is negative, and cosine is positive.
4. Apply the signs to the half angle values: Based on the signs determined in step 3, apply the appropriate sign to the values obtained from the half angle formulas. This will give you the correct value in the respective quadrant.
By considering the signs of the trigonometric functions and the quadrant of the original angle, you can determine the quadrant of the half angle values obtained from the half angle formulas.